Let `vec(M) and vec(L)` represent magnetic moment and angular momentum vectors for the electron in the above example. What is the sign of the dot product, `vec(M).vec(L)`. (positive, negative or zero) ?

Answer

To determine the sign of the dot product \(\vec{M} \cdot \vec{L}\) for the electron's magnetic moment \(\vec{M}\) and angular momentum \(\vec{L}\), we can follow these steps:

### Step 1: Understand the Definitions
- The **magnetic moment** \(\vec{M}\) for a revolving electron can be defined in terms of the current \(I\) and the area \(A\) it sweeps out:
\[ \vec{M} = I \cdot A \]
- The **angular momentum** \(\vec{L}\) of the electron is given by:
\[ \vec{L} = \vec{r} \times \vec{p} = \vec{r} \times (m\vec{v}) \]

### Step 2: Calculate the Current \(I\)
- The current \(I\) due to the electron moving in a circular path is given by:
\[ I = \frac{Q}{T} \]
where \(Q\) is the charge of the electron (which is negative, \(-e\)) and \(T\) is the time period of one complete revolution.

### Step 3: Determine the Time Period \(T\)
- The time period \(T\) for the electron moving in a circular path of radius \(r\) with velocity \(v\) is:
\[ T = \frac{2\pi r}{v} \]

### Step 4: Substitute \(I\) into the Magnetic Moment Equation
- Substituting \(I\) into the magnetic moment equation:
\[ \vec{M} = I \cdot A = \left(\frac{-e}{T}\right) \cdot \pi r^2 \]
After substituting \(T\):
\[ \vec{M} = \left(\frac{-e v}{2\pi r}\right) \cdot \pi r^2 = -\frac{e v r}{2} \]

### Step 5: Express Angular Momentum
- The angular momentum can be expressed as:
\[ \vec{L} = m \vec{v} \cdot r \]

### Step 6: Calculate the Dot Product \(\vec{M} \cdot \vec{L}\)
- The dot product \(\vec{M} \cdot \vec{L}\) is:
\[ \vec{M} \cdot \vec{L} = \left(-\frac{e}{2m} \vec{v} \cdot r\right) \cdot (m \vec{v} \cdot r) \]
Since \(\vec{M}\) and \(\vec{L}\) are in opposite directions, the dot product yields a negative value:
\[ \vec{M} \cdot \vec{L} < 0 \]

### Conclusion
The sign of the dot product \(\vec{M} \cdot \vec{L}\) is **negative**. This indicates that the magnetic moment and angular momentum vectors are in opposite directions.